Optimal. Leaf size=10 \[ \log \left (e^{-x}+e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 12, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2282, 446, 72} \[ \log \left (e^{2 x}+1\right )-x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 72
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int \frac {-e^{-x}+e^x}{e^{-x}+e^x} \, dx &=\operatorname {Subst}\left (\int \frac {-1+x^2}{x \left (1+x^2\right )} \, dx,x,e^x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x}{x (1+x)} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x}+\frac {2}{1+x}\right ) \, dx,x,e^{2 x}\right )\\ &=-x+\log \left (1+e^{2 x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 12, normalized size = 1.20 \[ \log \left (e^{2 x}+1\right )-x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 11, normalized size = 1.10 \[ -x + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 11, normalized size = 1.10 \[ -x + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 14, normalized size = 1.40 \[ \ln \left ({\mathrm e}^{2 x}+1\right )-\ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.88, size = 8, normalized size = 0.80 \[ \log \left (e^{\left (-x\right )} + e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.54, size = 11, normalized size = 1.10 \[ \ln \left ({\mathrm {e}}^{2\,x}+1\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 8, normalized size = 0.80 \[ - x + \log {\left (e^{2 x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________